2.1 Manifold Structure of P(H)
25
2-covariant tensor field x → Q x defining an isomorphism between T x P(H) and its
dual T
∗
x P(H) at any point x ∈ P(H):
v (∈ T x P(H)) → Q x (v, ·) ∈ T
∗
x P(H),
(2.1.17)
where the linear functional Q x (v, ·) : w (∈ T x P(H)) → Q x (v, w) ∈ R depends linearly on v, and for any F ∈ T
∗
x P(H) there is a unique v F ∈ T x P(H) such, that
F = Q x (v F , ·). Let the metric be given by
Q x (v, v) :=
2
x 2 (v x , v x ) =
2
x 2 v x
2
, v x := T x x (v).
(2.1.18)
Since v λx = λv x , the definition does not depend on the choice of 0 = x ∈ x in
the mapping x . The nondegeneracy is a consequence of the Riesz theorem applied
to the Hilbert space [x]
⊥ and analyticity is also straightforward. From the bilinearity
and symmetry we have
Q x (v, w) =
2
x 2 Re(v x , w x ), ∀v, w ∈ T x P(H).
(2.1.19)
It is possible to prove by straightforward calculations of lengths of differentiable
curves in P(H) (compare also [1, 262]):
2.1.10 Proposition. The metric Q from (2.1.19) endows P(H) with a distance
function d (calculated as the minimal length of differentiable curves joining two
points) different from d j , j = 1, 2; (2.1.2), (2.1.3). Both the distance functions d 1 , d 2
give (by differentiation) the metric Q from (2.1.19) on P(H).
2.2 Symplectic Structure
2.2.1 Let us define a complex structure J on P(H) induced by that of H. For each
x ∈ P(H) and v ∈ T x P(H), we define
J v := (T x x )
−1
◦ i ◦ (T x x )(v),
(2.2.1)
where i is the multiplication by the imaginary unit i ∈ C in the complex subspace
[x]
⊥
⊂ H. The definition (2.2.1) of J does not depend on the choice of x ∈ x.
Clearly: (J v) x = i v x . We define now a two-form on P(H) :
x (v, w) := Q x (v, J w), ∀x ∈ P(H), v, w ∈ T x P(H).
(2.2.2)